1.2 KiB
1.2 KiB
Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:
Triangle P3,n=n(n+1)/2 1, 3, 6, 10, 15, ...
Square P4,n=n2 1, 4, 9, 16, 25, ...
Pentagonal P5,n=n(3n?1)/2 1, 5, 12, 22, 35, ...
Hexagonal P6,n=n(2n?1) 1, 6, 15, 28, 45, ...
Heptagonal P7,n=n(5n?3)/2 1, 7, 18, 34, 55, ...
Octagonal P8,n=n(3n?2) 1, 8, 21, 40, 65, ...
The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.
- The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).
- Each polygonal type: triangle (
P(3,127)=8128), square (P(4,91)=8281), and pentagonal (P(5,44)=2882), is represented by a different number in the set. - This is the only set of 4-digit numbers with this property.
Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set.